A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Do all Stochastic matrix have a stationary probability vector?

I know that a stochastic matrix will have 1 as one of its eigenvalues. But do the stochastic matrices all have a stationary probability vector? Basically, could there be a case where the eigen ...
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14 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. If this is not acceptable - administrators please delete, ...
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a conceptual question on markov chain [duplicate]

Suppose $\{X_n,n\ge 0\}$ and $\{Y_n,n\ge0\}$ are two independent discrete-time markov chains (DTMC) with state space $S=\{0,1,2,\ldots\}$. Prove or give a counterexample to: $\{X_n+Y_n,n\ge 0\}$ is ...
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22 views

A credit model. Default time.

In a paper, I find the following situation: Let $(\Omega,\mathcal{G},\mathbb{Q})$ be a probability space. $\mathbb{Q}$ is supposed to be a risk neutral measure. Suppose that ...
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20 views

How to obtain accumulated counts of past events by time $t$?

Given $f: [0, \infty) \to \{0,1\}$, $f(t)$ represents whether there is an event occurring at time $t$. How can we obtain $g: [0,\infty) \to \mathbb{N}_0$ so that $g(t)$ represents the number of ...
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Showing that $(X_n)$ obeys the Markov Property. [on hold]

Consider a process $(X_n)_{n\geq0}$ where we define $X_0 = 0$ and for $n \geq 1$: $$X_n = X_{n-1} + Z_n$$ where $Z_n$ for $n \geq 1$ are independent random variables on $\{ -1, 1 \}$ with ...
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17 views

Compute distribution in Hidden Markov models

Let $Z_1, Z_2, ..., Z_n$ be the latent variables, and $X_1, X_2, ... X_n$ be the observed ones in a hidden markov models. Let's assume that the parameters of the hidden Markov models are known: the ...
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10 views

Density of intersection of sets with boundary condition

I would like to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{n}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) : ...
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20 views

What is the interpretation of $\nu(dy - x)$ where $\nu$ is a Lévy measure?

In a paper I am reading, it is seemingly suggested that, if $\nu(dx)$ is a Lévy measure, then the following holds for a function $f(x)$ which is smooth (and satsifies some integrability conditions): ...
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35 views

hitting times and stopping times

stopping times are always hitting times, but not the other way around. As an example of this, Last exit times are not stopping times as they depend on future information. the last exit time of $A$: ...
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45 views

Age distribution when meeting

I have a question regarding Poisson process. I will tell the story in the context of a player-monster game. Consider a player who is born at $t=0$. He will win the game if he can survive until ...
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28 views

Probability that a stochastic process is below a special random level

Given a stochastic process $x(t)$ over time $t \in [0,T]$, and a given (deterministic) $\tau$, where $0<\tau<T$, define a random variable $x^{*}$ as $$ x^{*} \triangleq \inf\bigg\{y: ...
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1answer
20 views

Weak convergence for composition of cadlag stochastic processes

Let $(X^n_t)_{t \geq 0}$ be a sequence of cadlag stochastic processes, that is $X^n$ is a random element in the Skorokhod space $D([0, \infty), \mathbb{R}$) for each $n \in \mathbb{N}$. Also for each ...
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13 views

Lognormal approximation of the sum of successive values of a lognormal process

I would like to use a lognormal process to approximate the successive values of another lognromal process. Let $X_t$ be a lognormal process. I would like to approximate $$ Y_t := \sum_{t=0}^T X_t $$ ...
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1answer
19 views

Predictability of $\int^t_0 f(X_s)\,\mathrm ds$ where $X$ is a Lévy process

Let $X_t$ be a Lévy process and $f(x)$ some smooth function. Under what conditions is $$ Y_t = \int^t_0 f(X_s)\,\mathrm ds$$ predictable? Not sure how to investigate this. It is clearly adapted, so ...
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1answer
16 views

Find $P(\eta_t=m)$, $m=0,1,2,\dots,$

Let $\epsilon_t$, $t=1,2,\dots$ independent random variables with $P(\epsilon_t=1)=p$ and $P(\epsilon_t=-1)=1-p$. If $\eta_0=0,\eta_t=\eta_{t-1}+\epsilon_t$ , $t=1,2,\dots$ where $\eta_t$ is ...
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13 views

Stopping of quadratic covariaton

I am given two local martingales $M$ and $N$ and a stopping time $\tau$. We work on a finite time interval $[0,T]$. I want to prove $$\langle M,N\rangle^{\tau}=\langle M^\tau,N\rangle$$ using the ...
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1answer
17 views

Expected value of distance between independent Brownian motions

Suppose $\{W^{(1)}_t, t\geq 0\}$ and {$W^{(2)}_t, t\geq 0\}$ are two independent Brownian motions. If I recall correctly, the distance between the two at a given time has the following property: ...
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41 views

one dimensional SDE with zero drift

I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, ...
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34 views

Measurability of marginal distributions of a random measurable function

For a probability space $(\Omega, \mathcal F, \mathsf P)$, let $X \colon \Omega \times [0,1] \to \mathbf R \colon (\omega, t) \mapsto X(\omega,t)$ be a random Borel function (i.e. an $(\mathcal ...
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37 views

Algebra behind Feynman-Kac theorem?

According to many sources, The Feynman-Kac theorem in Equation (1) below is the solution to Equation (3) - if X(t) follows a diffusion such as in (2). (Most Important) - Can someone show the algebra ...
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1answer
25 views

Covariance of Ornstein-Uhlenbeck process

$U(t)=e^{-\mu t}W(\frac{\sigma^2e^{2\mu t}}{2\mu})$. The problem is to find $Cov[U(t),U(t+s)]$. I used the identity, $W(\frac{\sigma^2e^{2\mu t}}{2\mu})=W(\frac{\sigma^2e^{2\mu t}e^{2\mu s}}{2\mu ...
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16 views

Stochastic variation in stockexchange, weather sciences [on hold]

As the Weierstrass continuous function has no derivative defined, its curvature or differential equations of the function is meaningless. Is that correct? Is there a definition of sufficiently ...
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1answer
70 views

Does this game make you arbitrarily rich with probability one?

We toss a coin. If it's heads we win $\$ 1$, otherwise we lose $ \$ 1$. Fix some large sum. Will we be winning this amount with probability one at some point? We assume that we have infinitely many ...
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Is there a Little law for a network of two connected queues?

From Patterson et al' Computer Organization and Design: Throughput and Response Time Do the following changes to a computer system increase throughput, decrease response time, or both? ...
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17 views

Stochastic optimal control : infinite horizon problem

Assume an investor has utility function $U(C_t)=\frac{C_t^\gamma}{\gamma}$. The investor wishes to consume some of their wealth at a rate $C_t$ per unit time, and invest in both risk-free bonds and a ...
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1answer
8 views

Common factors in ARIMA(p,d,q)

I have some concerns regarding interpreting ARIMA processes, A general ARIMA process is on the form $$ \phi(B)X_t = \theta(B)Z_t,\,\,Z_t\sim WN(0,\sigma^2)$$ For example if I have $$Y_t = ...
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2answers
50 views

piecewise weak convergence in $C[0,1]$

Let $P$ and the sequence $P_n$ be probability measures on $C[0,1]$ with the uniform metric. Fix $0<u<1$ and let $\Pi_1$ and $\Pi_2$ be the projections from $C[0,1]$ onto $C[0,u]$ and $C[u,1]$, ...
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What are the statistics of the discrete Fourier transform of a Bernoilli process?

The problem I would like to understand the statistics of the discrete Fourier transform of a sequence of uncorrelated events $\{x_n\}$ each of which takes the value $\pm1$ with probability $1/2$. In ...
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1answer
36 views

Show that if $\{X_n\}$ is a Markov Chain

Show that, if $\{X_n\}$ is a Markov Chain then $$P(X_n=j\mid X_k=l,X_m=i)=P(X_n=j\mid X_m=i),0\leq k<m<n$$ What I did is $$P(X_n=j\mid ...
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What is the Skewness of a Geometric Brownian Motion?

Consider a GBM : $$S(t) = S(0)\exp\left({(\mu-\frac{1}{2}\sigma^2) t + \sigma W_t}\right)$$ $$d\log S(t) = (\mu-\frac{1}{2}\sigma^2) t + \sigma dW_t$$ $$\frac{d S(t)}{S(t)} = \mu t + \sigma ...
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57 views
+50

How can I solve this stochastic system of equation?

$(B_1(t),B_2(t))$ is a 2-dimensional standard Brownian motion. $\alpha , \beta$ are constant. The system of equations is: $$dX_1(t)=X_2(t)dt+\alpha dB_1(t)\\dX_2(t)=-X_1(t)dt+\alpha dB_2(t)$$ I tried ...
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52 views

Disjoint increments of Poisson mixture process are memory-less

Let $N(t)$ be a Poisson mixture process: $$N(t) \sim (1-p) \cdot \text{Poiss}(\lambda_0 \cdot t) \: + \: p \cdot \text{Poiss}(\lambda_1 \cdot t),$$ where $p$ is fixed and $0<p<1.$ As we ...
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Measure of Sample Paths that Never Cross LIL Bound

Suppose that $X_i$ is an i.i.d. sequence of random variables, with $P(X_i=1)=P(X_i=0)=1/2$. Then $S_n = \sum_{i=1}^n$ is a zero mean random walk. From the Law of the Iterated Logarithm, for all ...
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20 views

Process Transition Algorithm

I have a process with 100 possible states and independent entities going through the process. All the Entities have been observed through a span of 5 years at the end of each month. When the ...
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15 views

Are some some particular subspaces of cadlag functions Polish?

Consider the space $D := D((0, \infty), \mathbb{N})$ of cadlag functions $f : (0, \infty) \to \mathbb{N}$ equipped with the Skorokhod $M_1$-topology. Then $D$ is Polish. Question 1: I want to check ...
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25 views

The definition of Random time. [closed]

I define a random time of a Martingale $ \lbrace Z_n: n \geq 1 \rbrace $ to be the random variable $ N $ for which there is a function $ f(Z_1, Z_2,..., Z_n) $ such that $ P \lbrace N=n | ...
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12 views

Exact solution to nonlinear backward SDE

I have read a paper about numerical SDE. After deriving the method, it uses the method to calculate the following nonlinear cases: $$\begin{cases} dX_t=ud\tau+\sigma ...
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15 views

Probability Law of Stochastic Process Definition

I am reading Probability and Stochastics by Çınlar, and am confused by the following definition in it: I must be missing something because this definition does not seem correct to me. For ...
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Can someone, please, suggest some books for Stochastic Processes with exercises?

Can someone, please, recommend me some books about Stochastic Processes,Martingales and Brownian Motion with many exercises? (I would be very happy if some of them are for beginners :D) Thank you!
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Can we have a Brownian motion under two different probability space?

Is it possible to construct a stochastic process $B_t$ such that $B_t$ is a Brownian under $(\Omega, F, P)$ and $B_t$ is a Brownian under $(\Omega, F, \hat{P})$? If not, how to argue that $P=\hat{P}$? ...
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Find a family of measures that satisfies the requirements for measurable selection

In chap 12 of Stoock and Varadhan Multidimensional diffusion processes in section 12.2 markov selections page 290 one reads I couldn't find an example that fit the conditions (a)-(d). One would ...
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13 views

Finding a function to use for Ito's Lemma

The original problem was to show the following stochastic process has a global solution: $$dx_i = x_i\left(b_i-\sum_{j=1}^4 a_{ij}x_j \right)dt+\sigma_ix_idW_t$$ To do so, they considered the ...
2
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Compute the Gibbs energy

I have a question about Gibbs distribution in Stochastic theory. In which, it defined a clique as a a subset $C$ in the whole image $\Omega$ if two different element of $C$ are neighbors. Figure 2 ...
2
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1answer
171 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
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27 views

What does it means of Normalization term of Gibbs distribution?

I am studying about Gibbs distribution concept and I am confusing about the term" normalization ". According to the Hammersley–Clifford theorem, an random $x$ can equivalently be characterized by a ...
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25 views

What is a Meyer Process?

Let $X$ be a square-integrable martingale. I am reading the following: Let $\langle X \rangle_t$ be a Meyer process, i.e. the unique predictable process with $\langle X \rangle_0=0$ and ...
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26 views

Book Recommendation for Studying Stochastic Optimization Problem with Almost Sure Constraint

I was begin to study the following type of stochastic optimization problem: Let $u(k)$ and $X(k)$ are discrete random variables for all $k =1,2,...,N$ and $f$ be any concave function and $g$ be ...
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24 views

How is the Laplace transform of the density of a specific point process computed?

I am trying to understand a little of this thesis by Anna Rudas. In particular the continuous model presented in Section 2.2.2. We are given a weight function $w: \mathbb{N} \rightarrow ...
2
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13 views

Differentiability of $E[1_{\tau > T} \mid X_t = x]$ where $X_t$ is a Lévy process

Let $X$ be a finite-variation Lévy process which starts at $X_0>0$, has positive drift, and has only downward jumps. Also define a stopping time $\tau := \inf(0\leq t \leq T: X_t<0)$, the first ...